Supersymmetric Yang-Mills, octonionic instantons and triholomorphic curves

TL;DR

This paper explores the deep connections between supersymmetric Yang-Mills theory, octonionic instantons, and triholomorphic curves, revealing how dimensional reductions of high-dimensional theories underpin these geometric correspondences.

Contribution

It demonstrates that the correspondence between octonionic instantons and triholomorphic curves arises from the dynamics of dimensional reductions of ten-dimensional supersymmetric Yang-Mills theory.

Findings

Shows the reduction of a 6D supersymmetric sigma model localizes on triholomorphic maps.

Interprets the topological sigma model as an adiabatic limit of 10D supersymmetric Yang-Mills.

Connects the moduli space of instantons to a cohomological theory in higher dimensions.

Abstract

In four-dimensional gauge theory there exists a well-known correspondence between instantons and holomorphic curves, and a similar correspondence exists between certain octonionic instantons and triholomorphic curves. We prove that this latter correspondence stems from the dynamics of various dimensional reductions of ten-dimensional supersymmetric Yang-Mills theory. More precisely we show that the dimensional reduction of the (5+1)-dimensional supersymmetric sigma model with hyperkaehler (but otherwise arbitrary) target X to a four-dimensional hyperkaehler manifold M is a topological sigma model localising on the space of triholomorphic maps M -> X (or hyperinstantons). When X is the moduli space M_K of instantons on a four-dimensional hyperkaehler manifold K, this theory has an interpretation in terms of supersymmetric gauge theory. In this case, the topological sigma model can be understood as an adiabatic limit of the dimensional reduction of ten-dimensional supersymmetric Yang-Mills on the eight-dimensional manifold M x K of holonomy Sp(1) x Sp(1) in Spin(7), which is a cohomological theory localising on the moduli space of octonionic instantons.